Matrix Quasi-tree Theorem

TL;DR

This paper generalizes the Matrix Tree Theorem to all embedded graphs by introducing symbolic skew-adjacency matrices and reduction maps, enabling enumeration of spanning quasi-trees in complex topological graph structures.

Contribution

It develops a comprehensive theory for embedded graphs, extending the Matrix Tree Theorem to quasi-trees using novel algebraic tools.

Findings

Polynomial encodes all spanning quasi-trees of a bouquet

Applicable to orientable and non-orientable embedded graphs

Provides a complete analogue of the Matrix Tree Theorem

Abstract

Building on prior work that established Matrix Quasi-tree Theorems for special embedded graphs, in this paper, we develop a comprehensive theory applicable to all embedded graphs. We introduce symbolic skew-adjacency matrices and reduction maps as key innovations, and prove that a specific polynomial derived from these matrices encodes all spanning quasi-trees of a bouquet. This result provides a complete analogue of the Matrix Tree Theorem for topological graph theory, with applications to quasi-tree enumeration in both orientable and non-orientable embedded graphs.